Marmari is a big beach, I found no problems kitesurfing there a few years ago, but it is the most crowded with many many many beachgoers. You'll have to walk a ways , but it is windier. You cannot
kite in the swim zone.
Further west is Mastichari, much less crowded, a little less of a walk to launch. cool little town.
This is where we stayed mostly.
Even further west is Kohilari, the last beach on Kos that faces north is the absolutely least crowded beach and really nice turquoise water.
Finally, all the way at the other end of the island going EAST, Kos town east end, Psalidi , is totally open. pebble stone beach, no sand
We had a car. Those are the basic kitesurf spots on Kos.